Question

The product of two consecutive odd integers is 99. Find the integers.

Short answer

Two consecutive odd integers whose product is 99 are 9, 11 and $-9,-11$.

Step-by-step solution

Step 1. Form a quadratic equation.

It is given that the product of two consecutive odd integers is 99.

Let $n$be the first odd integer and $n+2$is the second odd integer. Then,

$\begin{array}{rcl}n(n+2)& =& 99\\ n^2+2n-99& =& 0\end{array}$

Step 2. Solve the equation.

Split the middle term.

$\begin{array}{rcl}{n}^2+11n-9n-99& =& 0\\ n(n+11)-9(n+11)& =& 0\\ (n-9)(n+11)& =& 0\\ n& =& 9\text{and}n=-11\end{array}$

Step 3. Determine the second odd integer.

For $n=9$,

$\begin{array}{rcl}n+2& =& 9+2\\ & =& 11\end{array}$

For$n=-11$

$\begin{array}{rcl}n+2& =& -11+2\\ & =& -9\end{array}$

Step 4. Conclusion.

Two consecutive odd integers whose product is 99 are 9, 11 and $-9,-11$.